Acceptance-blind causal selection strictly extends standard sequential uniformization

Exact statement

For a regular source language A, finite causal selector covers are equivalent, up to explicit state overhead, to domain-covering synchronous multi-sequential subrelations of the accepting-lift relation R_(A,pi). For arbitrary nonregular sources the notions separate: under the identity projection every language has a one-state causal selector, while a standard sequential uniformizer can exist only when its required domain is regular.

StatusSelf-contained reduction proof with literature calibration and exhaustive finite audit

External reviewThe note explicitly calibrates against Jecker-Filiot multi-sequential relations and the sequential-uniformization literature. No specialist review of the FPRD acceptance-blind formulation is recorded.

Context

Causal selectors resemble sequential uniformizers but do not recognize the projected domain. T172 identifies the exact regular overlap and the strict nonregular gap before further selector theory is developed.

Hypotheses and scope

The projection is letter-to-letter between finite alphabets.
For the finite-family forward direction, the source language A is regular.
Transducers are deterministic, synchronous, and finite state.

Proof or evidence

Restricting a selector by a regular source DFA yields a partial sequential component with product states. Totalizing a partial component adds one sink and yields a causal selector. The identity projection of any nonregular language supplies the strict separation because sequential-transducer domains are regular.

Acceptance-blind uniformization bridge →

Verification notes

The saved checker covers 1,024 two-state source DFAs, four one-state selectors, 126,976 product-domain comparisons, 317,440 finite-cover checks, and nine partial one-state selector tables. Its equivalence counter repeats one predicate, and its totalization fixture is one-state; the written proof carries the general reduction.

Limitations

The result concerns synchronous letter-to-letter selection, not asynchronous uniformization.
For nonregular A, a successful selector's acceptance domain need not be regular, so the finite-family equivalence does not apply.
The theorem does not provide a synthesis procedure for nonregular source languages.
Uniformization and multi-sequentiality are established notions; no priority claim is made for them.

Open work

Determine which structural properties of nonregular source recognizers let acceptance-blind selectors be synthesized or ruled out effectively.